Three Tangent Circles – Geometry Problem With Detailed Solution

In this geometry problem, three circles are mutually tangent to each other and also tangent to a common line \( L \). The radii of circles \( A \) and \( B \) are given, and we determine the radius of circle \( C \).

Problem

Three circles are tangent to each other and to a line \( L \). The radius of circle \( A \) is \( 10 \text{ cm} \), and the radius of circle \( B \) is \( 8 \text{ cm} \). Find the radius of circle \( C \).

Three tangent circles geometry problem diagram

Step-by-Step Solution

Let the radii of the three circles be:

\[ a = 10, \quad b = 8, \quad c = ? \]

Draw perpendiculars from the centers of the circles to line \( L \). Applying the Pythagorean theorem to the appropriate right triangles allows us to determine horizontal distances between the centers.

Three tangent circles solution diagram

Step 1: Triangle \( BCB' \)

Using the Pythagorean theorem: \[ x^2 + (b - c)^2 = (b + c)^2 \] Expanding: \[ x^2 + b^2 - 2bc + c^2 = b^2 + 2bc + c^2 \] Simplifying: \[ x^2 = 4bc \] \[ x = 2\sqrt{bc} \]

Step 2: Triangle \( ACA' \)

\[ y^2 + (a - c)^2 = (a + c)^2 \] Expanding and simplifying: \[ y^2 = 4ac \] \[ y = 2\sqrt{ac} \]

Step 3: Triangle \( ABA'' \)

\[ z^2 + (a - b)^2 = (a + b)^2 \] Simplifying: \[ z^2 = 4ab \] \[ z = 2\sqrt{ab} \]

Step 4: Using the Geometric Relationship

Since the horizontal distance satisfies: \[ z = x + y \] Substitute the expressions: \[ 2\sqrt{ab} = 2\sqrt{bc} + 2\sqrt{ac} \] Divide both sides by 2: \[ \sqrt{ab} = \sqrt{bc} + \sqrt{ac} \] Factor \( \sqrt{c} \): \[ \sqrt{ab} = \sqrt{c}(\sqrt{b} + \sqrt{a}) \] Solve for \( \sqrt{c} \): \[ \sqrt{c} = \frac{\sqrt{ab}}{\sqrt{a} + \sqrt{b}} \] Square both sides: \[ c = \frac{ab}{(\sqrt{a} + \sqrt{b})^2} \]

Step 5: Substitute the Given Values

\[ c = \frac{10 \times 8}{(\sqrt{10} + \sqrt{8})^2} \] \[ c \approx 2.2 \text{ cm} \]

Final Answer

\[ \boxed{c \approx 2.2 \text{ cm}} \]

Therefore, the radius of circle \( C \) is approximately 2.2 cm.